Abstract

A new kind of partially coherent beam with non-conventional correlation function named generalized multi-Gaussian correlated Schell-model (GMGCSM) beam is proposed. The GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Furthermore, we carry out experimental generation of a GMGCSM beam of the first or second kind, and measure its focused intensity. Our experimental results verify theoretical predictions. The GMGCSM beam will be useful for free-space optical communications, material thermal processing, particle or atom trapping.

© 2014 Optical Society of America

1. Introduction

Since Gori et al. discussed the sufficient conditions for devising genuine correlation functions of scalar and electromagnetic partially coherent beams [1, 2], a variety of partially coherent beams with non-conventional correlation functions have been introduced [321]. Partially coherent beams with non-conventional correlation functions display many extraordinary properties, such as far-field flat-topped [47], four-beamlet array [8] and dark hollow (i.e., ring-shaped) beam profile formation [912], optical cage formation near the focal plane [13], self-focusing and a lateral shift of the intensity maximum [14, 15], far field radially polarization formation [16] and are useful in free-space optical communications, material thermal processing and particle trapping [13, 1721]. Multi-Gaussian correlated Schell-model (MGCSM) beam (i.e., multi-Gaussian Schell-model beam) whose degree of coherence (i.e., correlation function) is modeled by multi-Gaussian distribution was proposed by Korotkova et al. [46], and such beam has a flat-topped beam profile in the far field (or in the focal plane). Propagation properties of various MGCSM beams have been studied in detail [47, 20, 21], and it was found that MGCSM beams are less affected by turbulence than conventional Gaussian Schell-model beams and are useful in free-space optical communications. In this paper, a generalized multi-Gaussian correlated Schell-model (GMGCSM) beam whose degree of coherence is also modeled by multi-Gaussian distribution is proposed. The proposed GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Experimental generation of a GMGCSM beam of the first or second kind is reported.

2. Generalized multi-Gaussian correlated Schell-model beam: Theory

The statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function (MCF) in the space-time domain [22]. The MCF of the GMGCSM beam is defined as

J0(r1,r2)=exp[r12+r224σ02]γ(r1,r2),
where ri(xi,yi) is the position vector in the source plane, σ0 denotes the beam width, and the degree of coherence (DOC) γ(r1,r2) is expressed as
γ(r1,r2)=1Cαn=12Nm=12N(1)n+mAmnαBmnαexp[Bmnα(r1-r2)22δ02],
whereα=D,F, and δ0denotes the coherence width, N is the beam index. For the case ofα=D, Amnα,Bmnα,Cα take the following form
AmnD=(4N2n1)(4N2m1),BmnD=2mnm+n,
CD=m=12Nn=12N(1)n+mAmnDBmnD.
For the case of α=F, Amnα,Bmnα,Cα take the following form
AmnF=(2Nn)(2Nm),BmnD=2/(m+n),
CF=m=12Nn=12N(1)m+nAmnFBmnF.
The beam whose MCF is given by Eqs. (1)-(4) is termed GMGCSM beam of the first kind, and the beam whose MCF is given by Eqs. (1), (2), (5) and (6) is termed GMGCSM beam of the second kind. Both the DOC of the GMGCSM beam and the DOC of the earlier model of the MGCSM beam are expressed as a finite sum of Gaussian functions, while the way of the summation of Gaussian functions, the weight coefficients and the parameters in the exponentials of the GMGCSM beam are totally different from those of the MGCSM beam. The cross line of the DOC of the GMGCSM beam of the first kind is shown in Fig. 1(a) and the corresponding result of the GMGCSM beam of the second kind is shown in Fig. 1(b). One finds that the distribution of the DOC in Fig. 1(a) is different from that of the DOC in Fig. 2(b). Due to such difference, the GMGCSM beam of the first kind produces a dark hollow beam profile and the GMGCSM beam of the second kind produces a flat-topped beam profile in the focal plane (or in the far field) as shown later, and the radius of the dark hollow beam profile increases and the flat-topped beam profile becomes more flat as the beam index N increases.

 

Fig. 1 Cross lines (y1y2=0) of the DOC of (a) the GMGCSM beam of the first kind and (b) the GMGCSM beam of the second kind versus (x1x2)/δ0 for different values of the beam index N.

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Fig. 2 Normalized intensity distribution (cross lineρy=0) of the GMGCSM beam of the first kind (Figs. 2(a)-2(c)) or the second kind (Figs. 2(d)-2(f)) at several propagation distances in free space for different values of beam index N. The initial beam width and coherence length are set as σ0=1mmand δ0=0.2mm.

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In general, the GMGCSM beam can’t be used to describe the MGCSM beam by choosing suitable beam parameters, while the distribution of the DOC of the GMGCSM beam of the second kind is similar to that of the DOC of the MGCSM beam [4, 5]. The MGCSM beam is capable of producing flat-topped beam profile in the far field (or in the focal plane), while the GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the far field (or in the focal plane). In this way, the GMGCSM beam is more general.

The MCF of the partially coherent beam should satisfy the condition of nonnegative definiteness [1, 22] and can be written in the form [1]

J0(r1,r2)=I(v)H*(r1,v)H(r2,v)d2v,
where H is an arbitrary kernel, I is a non-negative function and the asterisk denotes the complex conjugate. Equation (7) can be expressed in the following alternative form
J0(r1,r2)=Ji(v1,v2)H*(r1,v1)H(r2,v2)d2v1d2v2.
where
Ji(v1,v2)=I(v1)I(v2)δ(v1v2).
Here δ denotes the Dirac function. One finds from Eqs. (8) and (9) that a partially coherent beam with prescribed correlation function can be generated from an incoherent source with MCF Ji(v1,v2)through propagation by choosing suitable expressions of I and H. Here I and H represent the intensity of the incoherent source and the response function of the optical path, respectively.

If we set H and I as follows:

H(r,v)=iλf1T(r)exp[iπλf1(v22rv)],
I(v)=1πω02[n=12N(1)n1(4N2n1)exp(v2/nω02)]2,
whereT(r)=exp(r2/4σ02),ω0denotes the beam width, after substituting Eqs. (10) and (11) into Eq. (8), it reduces to the MCF of a GMGCSM beam of the first kind withδ0=λf1/πω0. If we set H in the form of Eq. (10) and I as follows:
I(v)=1πω02[n=12N(1)n1(2Nn)exp(nv2ω02)]2,
after substituting Eqs. (10) and (12) into Eq. (8), it reduces to the MCF of a GMGCSM beam of the second kind with δ0=λf1/πω0. Thus, if the beam emitted from the incoherent source, whose intensity is given by Eq. (11) or (12), first propagates through free space at distance f1, then propagates through a thin lens with focal length f1 and a Gaussian amplitude filter (GAF) with transmission functionT(r), the transmitted beam from the GAF will become a GMGCSM beam of the first or second kind.

Applying Eqs. (1), (2) and the generalized Collins formula [23, 24], we obtain the following expression for the MCF of a GMGCSM beam in the output plane after propagating through a stigmatic ABCD optical system

J(ρ1,ρ2)=1Cαn=12Nm=12N(1)n+mAmnαBmnαΔmn2×exp(ρs22σ02Δmn2ρd22Ωmn2Δmn2+ikρsρdRmn).
where
ρs=ρ1+ρ22,ρd=ρ2ρ1,Ωmn2=14σ02+Bmnαδ02,
Δmn=A2+(Bkσ0Ωmn)2,Rmn=BΔmn2DΔmn2A.
Here ρi(ρix,ρiy) is the position vector in the output plane; A, B, C and D are the transfer matrix elements of the optical system; k=2π/λ is the wave number with λ being the wavelength. The intensity of the output GMGCSM beam is obtained asI(ρ)=J(ρ,ρ).

As a numerical example, we study the propagation properties the GMGCSM beam of the first or second kind in free space. The elements of the transfer matrix for free space of distance z read as A = 1, B = z, C = 0 and D = 1. Figure 2 shows the normalized intensity distribution (cross line ρy=0) of the GMGCSM beam of the first kind (Figs. 2(a)-2(c)) or the second kind (Figs. 2(d)-2(f)) at several propagation distances in free space for different values of beam index N with σ0=1mm, δ0=0.2mm and λ=632.8mm. One sees clearly that the intensity distribution of the GMGCSM beam of the first or second kind in the source (z = 0) has a Gaussian profile and is independent of the DOC (i.e., correlation function) and the beam index N. With the increase of the propagation distance, the effect of DOC on the intensity distribution of the GMGCSM beam gradually appears. The intensity distribution of the GMGCSM beam of the first or second kind turns to dark hollow (see Fig. 2(c)) or flat-topped (see Fig. 2(f)) profile in the far field, and the dark size or flatness of the beam spot increases as the beam index N increases.

3. Generalized multi-Gaussian correlated Schell-model beam: Experiment

Now we carry out experimental generation of the proposed GMGCSM beam of the first or second kind with N = 1. The most important part in our experiment is to construct an incoherent source whose MCF is given by Eq. (9) and intensity is given by Eq. (11) or Eq. (12). Part I of Fig. 3 shows our experimental setup for generating the GMGCSM beam of the first or second kind with N = 1. A linearly polarized Gaussian beam emitted from a He-Ne laser beam (λ633nm) first passes through a half-wave plate (HP1), then passes through a Mach-Zehnder interferometer (MZI). Two orthogonally polarized beams are superimposed together at the output of the MZI. Here HP1 is used to control the amplitudes of the x and y linearly polarized beams through rotating the HP1. The beam expander (BE) in MZI is used to control the width of the x linearly polarized beam. The electric field of the beam at the output of the MZI can be expressed as

E(v,0)=axexp(v2ω02)ex+ayexp(v2ω12)ey.
where ex and eydenote the unit vector along x- and y-directions, respectively, axand aydenote the amplitudes of the x and y components of the field. ω0and ω1denote the beam width of the x and y components of the field. In our experiment, we set ω1=2ω0through varying the BE, and we set ax=aythrough varying theHP1. The output beam from the MZI passes through a linearly polarizer (LP) whose transmission axis forms an angle θ with the x axis and a HP2which is used to rotate the polarization direction of the beam along x-direction. Then the intensity distribution of the beam from the HP2is given by

 

Fig. 3 Experimental setup for generating a GMGCSM beam, measuring the square of the modulus of its DOC and its focused intensity. HP1, HP2, half-wave plates; PBS1, PBS2, polarization beam splitters; BE, beam expander; LP, linear polarizer; RGGD, rotating ground-glass disk; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC, personal computer.

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I(v)=ax2|exp(v2ω02)cosθ+exp(v22ω02)sinθ|2.

If the transmission angle θ of LP equals toπ/4, Eq. (17) reduces to

I(v)=ax22|exp(v2ω02)exp(v22ω02)|2.
One finds that Eq. (18) has the same form with Eq. (11) for N = 1 except for the coefficientax2/2, which is not an important coefficient. If θ equals to0.35π, Eq. (17) reduces to
I(v)=5ax24|exp(v22ω02)12exp(2v22ω02)|2.
One finds that Eq. (19) has same form with Eq. (12) for N = 1 except for the coefficient5ax2/4. The transmitted beam from the HP2illuminates on a rotating ground-glass disk (RGGD), producing a partially coherent source with Gaussian statistics. If the diameter of the beam spot on the RGGD is much larger than the inhomogeneity scale of the ground glass [25] which is satisfied in our experiment, the generated partially coherent source can be regarded as an incoherent source whose MCF is given by Eq. (9) and intensity is given by Eq. (18) for θ=π/4 or Eq. (19) for θ=0.65π. After passing through the thin lens L1 and the GAF with T(r)=exp(r2/4σ02),the incoherent beam with prescribed intensity becomes a GMGCSM beam of the first kind for the case of θ=π/4 or the second kind for the case of θ=0.65π with N = 1.

Part II of Fig. 3 shows our experimental setup for measuring the square of the modulus of the DOC and the focused intensity of the generated GMGCSM beam. The generated beam from the GAF is split into two beams by the BS. The reflected beam passes through the thin lens L2 with focal length f2, and arrives at the CCD. Both distances from GAF to L2 and from L2 to CCD are 2f2 (i.e., 2 f-imaging system), so the DOC of the beam in the CCD plane is the same as that in the source plane (just behind the GAF). The output signal from the CCD is sent to a PC to measure the square of the modulus of the DOC. The detailed principle and the measuring process can be found in [3].

The transmitted beam from the BS passes through the thin lens L3 with focal length f3 = 10cm, then arrives at the BPA, which is used measure the intensity distribution. The distances from GAF to L3 and from L3 to BPA are f3 and z, respectively. The elements of the transfer matrix between GAF and BPA read as

A=1z/f3,B=f3,C=1/f3,D=0.

Figure 4 shows our experimental results of the square of the modulus of the DOC of (a) the generated GMGCSM beam of the first kind and (b) the generated GMGCSM beam of the second kind with N = 1. Through theoretical fit of the experimental data, we obtain δ0=0.054mmfor the generated GMGCSM beam of the first kind and δ0=0.035mmfor the generated GMGCSM beam of the second kind. One finds that the distribution of the square of the modulus of the DOC of the generated GMGCSM beam of the second kind is quite similar to Gaussian distribution, which is caused by the fact that the side lobe is too weak to be detected in our experiment, while the focusing properties of the generated GMGCSM beam is not influenced.

 

Fig. 4 Experimental results of the square of the modulus of the DOC of (a) the generated GMGCSM beam of the first kind and (b) the generated GMGCSM beam of the second kind with N = 1 versus x1 with x2 = y1 = y2 = 0. The solid line denotes the theoretical fit of the experimental results.

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Figures 5 and 6 show our experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the first kind and the generated GMGCSM beam of the second kind in the source plane and at two propagation distances after passing through the thin lens L3. For the convenience of comparison, the theoretical results calculated by Eq. (13) are also shown in Figs. 5 and 6. One finds from Figs. 5(a) and 5(d), Figs. 6(a) and 6(d) that the generated GMGCSM beam of the first kind has the same intensity distribution with that of the generated GMGCSM beam of the second kind in the source plane with σ0=1mm. From Figs. 5(b), 5(e), 5(c), 5(f), Figs. 6(b), 6(e), 6(c) and 6(f), one sees that the generated GMGCSM beam of the first kind and the generated GMGCSM beam of the second kind display quite different focusing properties, the former produces dark hollow beam profile while the latter produces flat-topped beam profile in the focal plane (or in the far field) due to different initial correlation functions. The focusing properties GMGCSM beam of the second kind is similar to that of the MGCSM beam proposed in [3, 4] which also displays flat-topped beam profile in the focal plane (or in the far field). Our experimental results are quite consistent with the theoretical predictions.

 

Fig. 5 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the first kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 4(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and Fig. 4(f) denote the theoretical results calculated by Eq. (13).

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Fig. 6 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the second kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 5(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and (f) denote the theoretical results calculated by Eq. (13).

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The experimental setup in this paper is only for generating GMGCSM beams with N = 1. In principle, the GMGCSM beams with higher values of N also can be generated for modifying the Mach-Zehnder interferometer in the experimental setup. Generation of the GMGCSM beams with N = 1 requires superposition of two Gaussian fields, while generation of the GMGCSM beams with N>1 requires superposition of 2N Gaussian fields in the interferometer. The interferometer should be carefully devised to realize the GMGCSM beam with higher values of N, and we leave this for future study.

4. Summary

We have introduced a new kind of partially coherent beam with non-conventional correlation function named GMGCSM beam, and derived its paraxial propagation formula. We have found that the GMGCSM beam of the first kind produces dark hollow beam profile and the GMGCSM beam of the second kind produces flat-topped beam profile in the focal plane (or in the far field). Furthermore, we have generated the GMGCSM beam of the first or second kind, and studied its focusing properties experimentally. Our experimental results have verified theoretical predictions. The proposed GMGCSM beam of the first kind will be useful for particle or atom trapping, where dark hollow beam spot is used to trap a particle [26] or atom [27]. The GMGCSM beam of the second kind will be useful in material thermal processing and free-space optical communications, where flat-topped beam spot is preferred [20, 21, 28].

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11474213&11404007&11274005&11104195, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Provsince under Grant No. 11KJB140007, Anhui Provincial Natural Science Foundation of China under Grant No.1408085QF112, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References and links

1. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]   [PubMed]  

2. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]  

3. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef]  

4. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef]   [PubMed]  

5. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]   [PubMed]  

6. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef]   [PubMed]  

7. Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014). [CrossRef]  

8. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef]   [PubMed]  

9. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef]   [PubMed]  

10. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef]   [PubMed]  

11. Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014). [CrossRef]   [PubMed]  

12. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). [CrossRef]   [PubMed]  

13. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014). [CrossRef]   [PubMed]  

14. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]   [PubMed]  

15. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef]   [PubMed]  

16. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]  

17. Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef]   [PubMed]  

18. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014). [CrossRef]   [PubMed]  

19. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef]   [PubMed]  

20. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]  

21. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]  

22. L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge, 1995).

23. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]  

24. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]   [PubMed]  

25. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]  

26. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]  

27. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998). [CrossRef]  

28. D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002). [CrossRef]  

References

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  1. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  2. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  3. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref]
  4. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  5. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  6. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  7. Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
    [Crossref]
  8. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [Crossref] [PubMed]
  9. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  10. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [Crossref] [PubMed]
  11. Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
    [Crossref] [PubMed]
  12. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
    [Crossref] [PubMed]
  13. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
    [Crossref] [PubMed]
  14. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  15. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013).
    [Crossref] [PubMed]
  16. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
    [Crossref]
  17. Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  18. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
    [Crossref] [PubMed]
  19. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref] [PubMed]
  20. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
    [Crossref]
  21. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
    [Crossref]
  22. L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge, 1995).
  23. S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [Crossref]
  24. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [Crossref] [PubMed]
  25. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
    [Crossref]
  26. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
    [Crossref]
  27. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
    [Crossref]
  28. D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
    [Crossref]

2014 (9)

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

2013 (6)

2012 (3)

2011 (1)

2010 (1)

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

2009 (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

2007 (1)

2002 (2)

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref] [PubMed]

1998 (1)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

1970 (1)

Cai, Y.

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref]

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref] [PubMed]

Chen, R.

Chen, Y.

Collins, S. A.

Coutts, D. W.

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Ding, B.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Du, S.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Gbur, G.

Gori, F.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Grimm, R.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Gu, Y.

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Korotkova, O.

Lajunen, H.

Liang, C.

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Lin, Q.

Liu, L.

Liu, X.

Manek, I.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Mei, Z.

Ovchinnikov, Y. B.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Santarsiero, M.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Shchepakina, E.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Suyama, T.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Tong, Z.

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Wu, G.

Yuan, Y.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Zhao, C.

Zhu, S.

IEEE J. Quantum Electron. (1)

D. W. Coutts, “Double-pass copper vapor laser master-oscillator power-ampli- fier systems: generation of flat-top focused beams for fiber coupling and percussion drilling,” IEEE J. Quantum Electron. 38(9), 1217–1224 (2002).
[Crossref]

J. Opt. (1)

Y. Zhang and Y. Cai, “Random source generating far field with elliptical flat-topped beam profile,” J. Opt. 16(7), 075704 (2014).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[Crossref]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Opt. Lett. (11)

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Phys. Rev. A (2)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Other (1)

L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge, 1995).

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Figures (6)

Fig. 1
Fig. 1 Cross lines ( y 1 y 2 = 0 ) of the DOC of (a) the GMGCSM beam of the first kind and (b) the GMGCSM beam of the second kind versus ( x 1 x 2 ) / δ 0 for different values of the beam index N.
Fig. 2
Fig. 2 Normalized intensity distribution (cross line ρ y = 0 ) of the GMGCSM beam of the first kind (Figs. 2(a)-2(c)) or the second kind (Figs. 2(d)-2(f)) at several propagation distances in free space for different values of beam index N. The initial beam width and coherence length are set as σ 0 = 1 mm and δ 0 = 0.2 mm .
Fig. 3
Fig. 3 Experimental setup for generating a GMGCSM beam, measuring the square of the modulus of its DOC and its focused intensity. HP1, HP2, half-wave plates; PBS1, PBS2, polarization beam splitters; BE, beam expander; LP, linear polarizer; RGGD, rotating ground-glass disk; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC, personal computer.
Fig. 4
Fig. 4 Experimental results of the square of the modulus of the DOC of (a) the generated GMGCSM beam of the first kind and (b) the generated GMGCSM beam of the second kind with N = 1 versus x1 with x2 = y1 = y2 = 0. The solid line denotes the theoretical fit of the experimental results.
Fig. 5
Fig. 5 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the first kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 4(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and Fig. 4(f) denote the theoretical results calculated by Eq. (13).
Fig. 6
Fig. 6 Experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the second kind in the source plane and at two propagation distances after passing through the thin lens L3. The solid line in Fig. 5(d) denotes Gaussian fit of the experimental results. The solid lines in Fig. 4(e) and (f) denote the theoretical results calculated by Eq. (13).

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

J 0 ( r 1 , r 2 ) = exp [ r 1 2 + r 2 2 4 σ 0 2 ] γ ( r 1 , r 2 ) ,
γ ( r 1 , r 2 ) = 1 C α n = 1 2 N m = 1 2 N ( 1 ) n + m A m n α B m n α exp [ B m n α ( r 1 - r 2 ) 2 2 δ 0 2 ] ,
A m n D = ( 4 N 2 n 1 ) ( 4 N 2 m 1 ) , B m n D = 2 m n m + n ,
C D = m = 1 2 N n = 1 2 N ( 1 ) n + m A m n D B m n D .
A m n F = ( 2 N n ) ( 2 N m ) , B m n D = 2 / ( m + n ) ,
C F = m = 1 2 N n = 1 2 N ( 1 ) m + n A m n F B m n F .
J 0 ( r 1 , r 2 ) = I ( v ) H * ( r 1 , v ) H ( r 2 , v ) d 2 v ,
J 0 ( r 1 , r 2 ) = J i ( v 1 , v 2 ) H * ( r 1 , v 1 ) H ( r 2 , v 2 ) d 2 v 1 d 2 v 2 .
J i ( v 1 , v 2 ) = I ( v 1 ) I ( v 2 ) δ ( v 1 v 2 ) .
H ( r , v ) = i λ f 1 T ( r ) exp [ i π λ f 1 ( v 2 2 r v ) ] ,
I ( v ) = 1 π ω 0 2 [ n = 1 2 N ( 1 ) n 1 ( 4 N 2 n 1 ) exp ( v 2 / n ω 0 2 ) ] 2 ,
I ( v ) = 1 π ω 0 2 [ n = 1 2 N ( 1 ) n 1 ( 2 N n ) exp ( n v 2 ω 0 2 ) ] 2 ,
J ( ρ 1 , ρ 2 ) = 1 C α n = 1 2 N m = 1 2 N ( 1 ) n + m A m n α B m n α Δ m n 2 × exp ( ρ s 2 2 σ 0 2 Δ m n 2 ρ d 2 2 Ω m n 2 Δ m n 2 + i k ρ s ρ d R m n ) .
ρ s = ρ 1 + ρ 2 2 , ρ d = ρ 2 ρ 1 , Ω m n 2 = 1 4 σ 0 2 + B m n α δ 0 2 ,
Δ m n = A 2 + ( B k σ 0 Ω m n ) 2 , R m n = B Δ m n 2 D Δ m n 2 A .
E ( v , 0 ) = a x exp ( v 2 ω 0 2 ) e x + a y exp ( v 2 ω 1 2 ) e y .
I ( v ) = a x 2 | exp ( v 2 ω 0 2 ) cos θ + exp ( v 2 2 ω 0 2 ) sin θ | 2 .
I ( v ) = a x 2 2 | exp ( v 2 ω 0 2 ) exp ( v 2 2 ω 0 2 ) | 2 .
I ( v ) = 5 a x 2 4 | exp ( v 2 2 ω 0 2 ) 1 2 exp ( 2 v 2 2 ω 0 2 ) | 2 .
A = 1 z / f 3 , B = f 3 , C = 1 / f 3 , D = 0.

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